De Rham-Kodaira s Theorem and Dual Gauge Transformations

Transcription

1 De Rham-Kodaira s Theorem and Dual Gauge Transformations arxiv:hep-th/ v1 29 Nov 2000 Hisashi ECHIGOYA and Tadashi MIYAZAKI Department of Physics, Science University of Tokyo, Kagurazaka, Shinjuku-ku, Tokyo , Japan Abstract A general action is proposed for the fields of q-dimensional differential form over the compact Riemannian manifold of arbitrary dimensions. Mathematical tools are based on the well-known de Rham-Kodaira decomposing theorem on harmonic integral. A field-theoretic action for strings, p-branes and high-spin fields is naturally derived. We also have, naturally, the generalized Maxwell equations with an electromagnetic and monopole current on a curved spacetime. A new type of gauge transformations (dual gauge transformations) plays an essential role for coboundary q-forms. PACS number(s): Ky, De, Kk 1

2 1 Introduction It goes without saying that field theories play a central role in drawing a particle picture. They are especially important to explore a way to construct a theoretical view on a curved space-time (of more than four dimensions). Recently-developed theories of strings[1] and membranes[2], as well as those of two-dimensional gravity[3], go along this way. If one makes a complete picture with a general action, one may have a clear understanding about why the fundamental structure is ether of one dimension (a string), excluding other extended structures of two or more dimensions, or of other definite dimensions. The first purpose of this paper is to obtain a general action for the fields of q- dimensional differential forms (q-forms) on a general curved space-time. In such a way can we deal not only with strings and p-branes (p-dimensional extended objects), but also with vector and tensor fields as assigned over each point of a compact Riemannian manifold (e.g., a sphere or a torus of general dimensions). Our next aim is, as a result of this treatment, to generalize the conventional Maxwell theory to that on the curved space-time of arbitrary dimensions. Our method is based on the mathematical theory having been developed by de Rham and Kodaira[4]. In the theory of harmonic integrals the elegant theorem, having been now crowned with the names of the two brilliant mathematicians, says that an arbitrary differential form consists of three parts: a harmonic form, a d-boundary and a δ-boundary. With this theorem we have an electromagnetic field coming from the d-boundary, whereas a magnetic monopole field from the δ-boundary. We are thus to have a generalized Maxwell theory with an electric charge and a magnetic monopole on an arbitrary-dimensional curved space-time. Assigning a δ-boundary to a point on the curved space-time, we have a new kind of gauge freedom due to the nilpotency of the coboundary operator. Lastly, we comment on the possibility of the case where there could simultaneously exist a matter field and a new gauge field interacting together and invariantly under 2

3 the afore-mentioned new type of gauge transformations. In this paper we proceed to construct a field theory by taking various concrete examples. Section 2 treats an algebraic method for obtaining a general action. Sections 3 to 6 are devoted to concrete examples. In Sect.7 we comment on the case of interacting matter fields with a new gauge field. Two often-used mathematical formulas are listed in Appendix. We hope the method developed here will become one of the steps which one makes forward to construct the field theory of all extended objects strings and p-branes with or without spin degrees of freedom based on algebraic geometry. 2 A general action with q-forms Let us start with a Riemannian manifold M n, where we, observers, live, and with a submanifold M m, where particles live (n,m : dimensions of the manifolds; n m). Both M n and M m are supposed to be compact compact only because mathematicians construct a beautiful theory of harmonic forms over compact spaces, and de Rham-Kodaira s theorem or Hodge s theorem has not yet been proven with respect to the differential forms over non-compact spaces. We will admit the space M m of a particle to be a submanifold of M n. For instance, M m may be a circle or a sphere within an n-dimensional (compact) space M n. The local coordinatesystems of M n and M m shall be denoted by (x µ ) and(u i ), respectively [µ = 1,2,...,n;i = 1,2,...,m][5]. A point ( u 1,u 2,...,u m ) of Mm is, at the same time, a point of M n, so that it is also expressed by x µ = x µ (u i ). In a conventional quantum field theory, point particles, scalar fields, vector or higher-rank tensor fields, or spinor fields are attributed to each point of Mm. In this view we are to assign a q-dimensional differential form (q-form) F (q) to each point of Mm, which is expressed, as mentioned above, by thelocal coordinate(u 1,u 2,...,u m ) orby x µ = x µ (u i ). Physical objects 3

4 point particles, strings or electromagnetic fields should be identified with these q-forms. We then make an action with F (q). One of the candidates for the action S is (F (q),f (q) ) M m F(q) F (q), where means Hodge s star operator transforming a q-form into an (m q)-form. Expressed with respect to an orthonormal basis ω 1,ω 2,...,ω m, it is defined by the relation ( ) m (ω i1 ω i2... ω iq ) = ω (m q)! i 1... i q j 1... j j1 ω j2... ω jm q, m q (2.1) where (...) denotes the signature (±) of the permutation and the summation convention over repeated indices is, here and hereafter, always implied. The inner product (F (q),f (q) ) is a scalar and shares a property of scalarity with the action S. Let us, therefore, admit the action S to be proportional to (F (q),f (q) ) and investigate each case that we confront with in the conventional theoretical physics. Thus we put S = (F (q),f (q) ) = S M m = Ldu 1 du 2... du m, (2.2) M m S F (q) F (q) = Ldu 1 du 2... du m. Here S is an action form, but we will sometimes call it by the same name action. L is interpreted as a Lagrangian density. According to the well-known de Rham-Kodaira theorem, an arbitrary q-form decomposes into three mutually orthogonal q-forms: F (q) = F (q) I +F (q) II +F (q) III, (2.3) where F (q) I is a harmonic form, meaning[6] df (q) I = δf (q) I = 0, (2.4) and F (q) II is a d-boundary, and F (q) III is a δ-boundary (coboundary). Here δ is Hodge s adjointoperator, whichimpliesδ = ( 1) m(q 1)+1 d whenoperatedtoq-formsoverthe 4

5 m-dimensional space. There exist, therefore, a (q 1)-form A (q 1) II A (q+1) III, such that The action S is (proportional to) (F (q),f (q) ) ; and a (q+1)-form F (q) II = da (q 1) II ; F (q) III = δa (q+1) III. (2.5) S (F (q),f (q) ) = (F (q) I,F (q) I )+(A (q 1) II,δdA (q 1) II )+(A (q+1) III,dδA (q+1) III ), (2.6) S S = Ldu 1 du 2... du m. M m The physical meaning of Eq.(2.6) is whatever we want to discuss in this paper and will be described in detail from now on. 3 Point particles, strings and p-branes We first assign F (0) = 1 to a point (u 1,...,u m ) of the submanifold M m, and we always make use of the relative (induced) metric ḡ ij for Mm (so that the intrinsic metric of the submanifold is irrelevant). ḡ ij xµ (u) u i x ν (u) u j g µν, (3.1) where g µν is a metric of the Riemannian space M n. Since the volume element dv ω 1 ω 2... ω m is expressed, with respect to the local coordinate (u i ), as dv = ḡ du 1 du 2... du m = 1, (3.2) we immediately find (F (0),F (0) ḡ ) = du 1 du 2... du m, (3.3) M m with ḡ = det(ḡ ij ). 5

6 When n = 4 and m = 1, we have ( means d/du 1 ), hence ḡ = g µν dx µ du 1 dx ν du 1 = g µνẋ µ ẋ ν, (3.4) (F,F) = M 1 ds, (3.5) ds 2 = g µν ẋ µ ẋ ν (du 1 ) 2 = g µν dx µ dx ν, which indicates that (F,F) is a conventional action (up to a constant) for a point particle in a 4-dimensional curved space, with u 1, interpreted as a proper time. with On the contrary, if we treat a submanifold M 2, Eq.(3.3) becomes (F (0),F (0) ḡ ) = du 1 du 2, (3.6) M 2 ḡ = det( xµ u i x ν u jg µν), (3.7) which is just the Nambu-Goto action in a curved space (with u 1 = τ and u 2 = σ in the conventional notation). There, and here, the determinant ḡ of an induced metric plays an essential role. If we confront with an arbitrary submanifold M p+1 (p : an arbitrary integer n 1), we are to have a p-brane, whose action is nothing but that given by Eq.(3.3) with m = p+1. Let us discuss the transformation property of the action or Lagrangian density. The transformation of Mm into M m without changing M n [7] means reparametrization. u i u i, x µ (u i ) x µ (u i ) = x µ (u i ). (3.8) By this the volume element Eq.(3.2) does not change, so that our Lagrangian (density) for the p-brane is trivially invariant under the reparametrization. If we convert M n into M n without changing M m, a general coordinate transformation x µ (u i ) x µ (u i ) (3.9) 6

7 is induced, under which ḡ ij does not change, because of the transformation property of the metric g µν. Our action is trivially invariant also for this general coordinate transformation. If we transform M m and M n simultaneously, i.e., u i u i, x µ (u i ) x µ (u i ), (3.10) we donot have anequality x µ (u i ) = x µ (u i ). This typeof transformationsisexamined, as an example, for n = 3 and m = 2 as follows. Let us take M 3 = R 3 (compactified), and M 2 = S 2 (2-dimensional surface of a sphere) whose local coordinate system is (u 1,u 2 ). A point of S 2 is expressed by (u 1,u 2 ), but it is at the same time a point (x 1,x 2,x 3 ) of R 3. We give the relation between the two coordinate systems by the stereographic projection: x 1 = x 2 = 2r 2 u 1 (u 1 ) 2 +(u 2 ) 2 +r, 2 2r 2 u 2 (u 1 ) 2 +(u 2 ) 2 +r, (3.11) 2 x 3 = r[r2 (u 1 ) 2 (u 2 ) 2 ] (u 1 ) 2 +(u 2 ) 2 +r 2, where r is the radius of the sphere defining S 2. The transformation (u 1,u 2 ) (u 1,u 2 ) induces the transformation (x 1,x 2,x 3 ) (x 1,x 2,x 3 ), and vice versa. The definition of the metric g µν for M n and the induced one ḡ ij for M m tells us and hence so that we have g µν (x ) = xρ x µ x δ x νg ρδ(x), (3.12) ḡ ij(u ) = uk u i u l u jḡkl(u), (3.13) ḡ (u )du 1... du m = ḡ(u)du 1... du m, (3.14) 7

8 meaning the invariance of the action. 4 Scalar fields Now we consider the case where a scalar field φ(x µ (u i )) is assigned to each point x µ (u i ). From now on we regard every quantity as that given over the subspace M m, hence we will write the field simply as φ(u i ) or φ(u) instead of φ(x µ (u i )), etc. An arbitrary 0-form a scalar field decomposes into two parts: F (0) = F (0) I +F (0) III. (4.1) F (0) I is given by F (0) I = φ(u), (4.2) with which we obtain (F (0) I,F (0) I ) = φ 2 (u)dv, (4.3) meaning a mass term of a scalar field. F (0) III is composed, on the contrary, of a δ -boundary of a 1-form: F (0) III = δa (1), A (1) = A i du i. (4.4) Hence we have where, as usual, F (0) III = k ( ḡa k ) ḡḡ 11 ḡ 22...ḡ mm, (4.5) A k = ḡ kl A l and k = uk, (4.6) 8

9 and (ḡ ij ) is the inverse of (ḡ ij ). In a special case, where we work out with a flat space and an orthonormal basis, i.e., we have a simple form by which the action form S becomes This is the kinetic term of the k-vector field A k. The gauge transformation exists for this field: ḡ ij = δ ij and du i = ω i, (4.7) F (0) III = k A k, (4.8) S = ( k A k ) 2 dv. (4.9) A (1) Ã(1) = A (1) +δa (2), In components, it is written as ( 1 h l l à h =A h + m 1 2(m 2)! i 1 i 2 j 1... j m 2 A (2) = 1 2 A i 1 i 2 du i 1 du i 2. (4.10) ) ( ḡa i 1 i 2 ) u k ḡḡ kl1ḡ l 2j 1...ḡ l m 1j m 2.(4.11) One can further calculate, if one wants to, to have a beautiful form: à i = A i 1 ( ) j1 j 2 ḡ kl D 2 k i l A j1 j 2, where Γ i jk D l A j1 j 2 = A j 1 j 2 u l A kj2 Γ k j 1 l A j 1 kγ k j 2 l, (4.12) is the well-known affine connection. Γ i jk = 2ḡil 1 ( ḡ jl u + ḡ lk k u ḡ jk ). (4.13) j ul Note that our fundamental fields are the A i, and the gauge transformation is obtained with the A i1 i 2 of the rank higher by one than the former. This is, of course, due to the nilpotency of δ, δ 2 = 0, and typical of our new type of formulation. Let us call, here and hereafter, that new kind of gauge transformations dual gauge transformations. 9

10 5 Vector fields When a 1-form F (1) is assigned to each point of Mm, we have F (1) = F (1) I +F (1) II +F (1) III. (5.1) First we will see the contribution of F (1) I to the action, which is harmonic. Writing as we immediately have an action (form) F (1) I = F i du i, (5.2) S I = F (1) I F (1) I = F i F i ḡ du 1... du m (5.3) The contribution of the d-boundary is calculated in the same way. Putting we have the action F (1) II = da (0), (5.4) S II = ḡ ij i A (0) j A (0) ḡ du 1... du m, (5.5) which expresses a massless scalar particle A (0). Freedom of the choice of gauges does not here appear. The contribution of the δ-boundary is, on the contrary, rather complicated in calculation. If we put F (1) III = δa (2), A (2) = 1 2 A i 1 i 2 du i 1 du i 2, (5.6) F (1) III = F i du i, we have ( ) h l1 l F h = 2... l m 1 i 1 i 2 j 1... j m 2 u k( ḡa i 1i 2 ) ḡḡ kl1ḡj 1l 2...ḡ j m 2l m 1 = 1 ( ) j1 j 2 ḡ kl D 2 k h l A j1 j 2, (5.7) 10

11 with D l, defined in Eq.(4.12)[8]. The action is S III = F i F i ḡ du 1... du m. (5.8) The dual gauge transformation is given in this case by which trivially leads to the relation A (2) Ã(2) = A (2) +δa (3), A (3) = 1 3! A i 1 i 2 i 3 du i 1 du i 2 du i 3, (5.9) F (1) III = δa (2) = δã(2). (5.10) When expressed in components, it is written as ( 1 i1 i à h1 h 2 = A h1 h 2 2 i 3 j 1... j m 3 3!(m 3)! h 1 h 2 l l m 2 ) u k( ḡa i 1i 2 i 3 ) ḡḡ kl1ḡ j 1l 2...ḡ j m 3l m 2, (5.11) where, of course, the components with superscript are related to those with subscript in a conventional manner, as has been described repeatedly. A i 1i 2 i 3 = ḡ i 1j1ḡ i 2j2ḡ i 3j 3 A j1 j 2 j 3. (5.12) We finally express Eq.(5.11) in an elegant form. à h1 h 2 = A h1 h 2 1 ( j1 j 2 j 3 3! k h 1 h 2 ) ḡ kl D l A j1 j 2 j 3, D l A j1 j 2 j 3 = A j 1 j 2 j 3 u l A kj2 j 3 Γ k j 1 l A j1 kj 3 Γ k j 2 l A j1 j 2 kγ k j 3 l. (5.13) Especially when the space-time is flat and one takes an orthonormal reference frame, one has F i = 1 2 ( k i i 1 i 2 ) A i 1 i 2 which further reduces to a familiar form for m = 4: F i = k A ik, u k, (5.14) S = k A ik l A il dv. (5.15) 11

12 The dual gauge transformation becomes in this case à i1 i 2 = A i1 i 2 k A i1 i 2 k. (5.16) Needless to say, the total action comes from adding S I,S II and S III. A new type of gauge transformations Eq.(5.13) appears, due to the coboundary property of F (1) III. 6 Tensor fields Now we come to the case where a 2-form is assigned to each point of Mm, the case of which is most useful and attractive for future development. A 2-form decomposes, as usual, into the following three: F (2) = F (2) I +F (2) II +F (2) III. (6.1) The harmonic form F (2) I is written with the components A ij as follows: F (2) I = 1 2 A i 1 i 2 du i 1 du i 2, (6.2) from which we have S I = F (2) I F (2) I = 1 2 A i 1 i 2 A i 1i 2 ḡ du 1... du m. (6.3) The contribution ofthed-boundaryisexpressed withour fundamental 1-formA (1). This further reduces, when written in components, to a familiar relation F (2) II = da (1). (6.4) F (2) II = 1 2 F i 1 i 2 du i 1 du i 2, A (1) = A i du i, (6.5) F ij = i A j j A i, (6.6) 12

13 which shows that F ij is a field-strength. The gauge transformation here is given by Namely, it is expressed in components as A (1) Ã(1) = A (1) +da (0). (6.7) Ã i = A i + i A(u), (6.8) with A(u), an arbitrary scalar function, which is a familiar form in the conventional Maxwell electromagnetic theory. The invariance of the contribution to F (2) II owes selfevidently, to the nilpotency d 2 = 0. Let us now add a source term 2(A (1),J (1) ) to the action with J (1), a source of one -form. Then we have the equation of motion from Hamilton s principle of least action: In component it is written as follows: 1 ( 1 h l1 l 2... l m 1 2(m 2)! i 1 i 2 j 1... j m 2 δf (2) II = δda (1) = J (1). (6.9) ) u k( ḡf i 1i 2 ) ḡ ḡ l 1kḡ l 2j 1...ḡ l m 1j m 2 = J h. (6.10) After some lengthy calculations we finally have the following beautiful form. 1 ( ) i1 i 2 ḡ jl D 2 j h l F i1 i 2 = J h. (6.11) ThecovariantderivativeD l isgivenineq.(4.12). Equation(6.10)or(6.11)becomes simple for the f lat m-dimensional space, expressed in an orthonormal basis. F ij, j = J i (6.12) This is nothing but the Maxwell equation in an m-dimensional space, with J i, interpreted as an electromagnetic current density. One therefore finds that Eq.(6.9) or (6.11) is the generalized Maxwell equation in the curved m-dimensional space. Here we note that the invariance of the action under the gauge transformtion (6.7) or (6.8) is evident as long as the eqation for the current δj (1) = 0 (6.13) 13

14 holds. In case of the flat space with an orthonormal basis, this reduces to the usual form of the conservation of current i J i = 0. Now comes the contribution of the δ-boundary: where A (3) is a 3-form. Expressed, as usual, in components F (2) III = δa (3), (6.14) F (2) III = 1 2 F i 1 i 2 du i 1 du i 2, A (3) = 1 6 A i 1 i 2 i 3 du i 1 du i 2 du i 3, (6.15) Eq.(6.14) leads us to ( ) 1 h1 h F h1 h 2 = 2 l l m 2 6(m 3)! i 1 i 2 i 3 j 1... j m 3 u k( ḡa i 1i 2 i 3 ) ḡ ḡ l 1kḡ l 2j 1...ḡ l m 2j m 3. (6.16) Along the same line already mentioned repeatedly we further have F i1 i 2 = 1 ( ) j1 j 2 j 3 ḡ kl D 6 k i 1 i l A j1 j 2 j 3, (6.17) 2 with the covariant derivative D l A j1 j 2 j 3, defined in Eq.(5.13). In the same way as in the case of the d-boundary, we add a source term 2(A (3), K (m 3) ) to the action (6.3). The variation of A (3) gives us the following equation of motion: df (2) III = dδa (3) = K (m 3), K (m 3) 1 = (m 3)! K i 1 i 2...i m 3 du i 1... du i m 3, (6.18) One has the relation between the components of F (2) III and K (m 3) : ( ) m ḡk j F i1 i 2,i 3 +F i2 i 3,i 1 +F i3 i 1,i 2 = 1...j m 3, (m 3)! j 1... j m 3 i 1 i 2 i 3 (6.19) where F i1 i 2,i 3 F i1 i 2 / u i 3, etc.. If our space-time M m is flat and the dimension is m = 4, these expressions reduce to a familiar form. F µν = ρ A µνρ, F µν, ν = K µ, (6.20) 14

15 where F µν = 1 2 ǫ µνρσf ρσ, K (1) = K µ du µ. (6.21) Equations (6.20) and (6.21) tell us that K µ is a magnetic monopole current[10]. The dual gauge transformation is, in this case, given by A (3) Ã(3) = A (3) +δa (4). (6.22) In components is it written as à h1 h 2 h 3 = A h1 h 2 h 3 + ( 1 h1 h 2 h 3 l l m 3 4!(m 4)! i 1 i 2 i 3 i 4 j 1... j m 4 u k( ḡa i 1i 2 i 3 i 4 ) ḡ ḡ l 1kḡ l 2j 1...ḡ l m 3j m 4, (6.23) ) which one can further rewrite as follows: à i1 i 2 i 3 = A i1 i 2 i ( j1 j 2 j 3 j 4 4! k i 1 i 2 i 3 ) g kl D l A j1 j 2 j 3 j 4, D l A j1 j 2 j 3 j 4 = A j 1 j 2 j 3 j 4 u l A kj2 j 3 j 4 Γ k j 1 l A j 1 kj 3 j 4 Γ k j 2 l A j 1 j 2 kj 4 Γ k j 3 l A j 1 j 2 j 3 kγ k j 4 l. (6.24) The invariance of the action under the dual gauge transformation (6.22) is assured for the current K (m 3) that satisfies dk (m 3) = 0. (6.25) The action form S III = F (2) III F (2) III can be, of course, calculated along the same line already mentioned. And the total action S is S = S I +S II +S III 2(A (1),J (1) )+2(A (3), K (m 3) ) (6.26) 15

16 7 Comments and discussions We have taken, up to now, the position that we only have a gauge field (or a scalar Field) (q-form F (q) ) as a fundamental field. Here the question arises as to whether there can simultaneously exist a matter filed and a gauge field at the outset, both of which, together, interact with each other gauge-invariantly. This viewpoint is conventional, but a dual gauge transformation should be introduced if one has a well-defined δ-boundary. Unfortunately, we are led to a very restricted way of treating. As a simple example we manipulate an (m 2)-form F (m 2) III (δ-boundary) and a two-component real field φ A (u) [A = 1,2], assigned to each point of the manifold M m, i.e., F (m 2) III = δa (m 1), A (m 1) 1 = (m 1)! A i 1 i m 1 du i 1 du i m 1. (7.1) The dual gauge transformation is given by A (m 1) Ã(m 1) = A (m 1) +δa (m). (7.2) In components we have à i1 i m 1 ( 1 2 m = A i1 i m 1 k i 1 i m 1 ) ḡ kl D l A 12 m, D l A 12 m = A 12 m u l = A 12 m u l A 12 m Γ k kl 1 2 A 12 m lnḡ. (7.3) ul With these A i1 i m 1 we introduce a dual one-form B (1) whose components are B j as follows. B (1) A (m 1), ( 1 1 m 1 m B j = (m 1)! i 1 i m 1 j ) ḡa i 1 i m 1. (7.4) 16

17 The field strength B ij is given by B ij i B j j B i. (7.5) The dual gauge transformation reduces to a conventional form with this dual one-form; B k = B k + k λ, (7.6) where λ, a scalar dual to A i 1 i m, is ( 1 m 1 m ) ḡa i 1 i m 1i m The δ-boundary F (m 2) III λ(u) 1 m! i 1 i m 1 i m = ḡa 12 m. (7.7) F (m 2) III is calculated with this B (1) to be ( ) m = 2(m 2)! h 1 h 2 l 1 l m 2 ḡb kj ḡ kh1ḡ jh 2 du l 1 du l m 2. (7.8) The local U(1) gauge transformation for the matter field φ A (u) is obtained, with λ(u) now infinitesimal, Here we have the covariant derivative for φ A (u); ˆδφ A (u) = T A Bφ B (u)λ(u), ( ) 0 1 T =. (7.9) 1 0 i φ A (u) = i φ A (u) T A Bφ B (u)b i (u), (7.10) and we immediately have the covariance of i φ A (u); ˆδ i φ A (u) = T A B( i φ B (u))λ(u). (7.11) The total Lagrangian density is L tot = L (m 2) gauge +L matter, L gauge (m 2) = 2 ḡḡ 1 i 1 j1ḡ i 2j 2 B i1 i 2 B j1 j 2, L matter = 1 ḡ k φ A k φ A 1 ḡµ 2 φ A φ A, (7.12)

18 with µ, the mass of the matter field, and we give the Lagrangian for the gauge-field sector a minus sign, in order to have a positive energy. The equations of motion for the matter field φ A (u) and the gauge field B k (u)are, respectively, ḡbk k φ B T B A + ḡµ 2 φ A + k ( ḡ k φ A ) = 0, 2 l ( ḡb kl ) ḡ k φ A T A Bφ B = 0. (7.13) It goes without saying that the conserved Noether current exists for our U(1) gauge transformation. One wonders, here, that nothing differs in gauge transformation for δ- boundary from for the conventional d-boundary. The essential point is that, for and only for q = m 2, the dual gauge transformation reduces to an ordinary gauge transformation according as the (m 1)-form A i 1 i m 1 dually transforms to the vector ḡb k. In this case F (m 2) II becomes: A (m 3) II (d-boundary) = da (m 3) II, and the gauge transformation of A (m 3) II Ã(m 3) II = A (m 3) II +da (m 4) II. So as this gauge transformation be conventional, we must have m = 4. Hence we have the fact that in case of our spacetime being dimensional, we have both electromagnetic and monopole currents as well as the matter field. Now comes the conclusion. The q-form formulation over the compact Riemannian manifold leads us to the world where both electromagnetic and monopole currents exist. The mathematical tool we adopt is based on the de Rham-Kodaira decomposing theorem of harmonic forms. Higher-rank q-form endows a particle with an intrinsic degree of freedom (integer sign). In case of q = m 2, we are able to introduce both the matter field and dual gauge field (δ-boundary) from the beginning. For m = 4 and q = 2, we can start with three kinds of fields: Electromagnetic fields (d-boundary), dual fields (δ-boundary) and matter fields over the curved space-time. The last fields are coupled with the former two fields; the way of coupling is gauge invariant and dual-gauge invariant. 18

19 Acknowledgement One of the authors (H.E) thanks Iwanami Fūjukai for financial support. 19

20 A Hodge s star operator AsdefinedbyEq.(2.1), Hodge sstaroperator isanisomorphismofh q (linerspace of q-forms) into H m q. Here, in this appendix, we only write down two important formulas which we frequently use in calculation in Sects.4 to 7. For an arbitrary q-form ϕ = 1 q! ϕ i 1 i 2...i q du i 1 du i 2... du iq, (A.1) we have where ϕ = ( m (m q)!q! i 1... i q j 1... j m q ) ḡ ϕ i 1...i q du j 1... du j m q, (A.2) ϕ i 1...i q = ḡ i 1l 1...ḡ iqlq ϕ l1...l q, (A.3) with ḡ ij, the metric tensor. As for a basis of H q, we have (du k 1... du kq ) = ( m (m q)! i 1... i q j 1... j m q ḡ ḡ i 1k 1...ḡ iqkq du j 1... du j m q. (A.4) ) Note that a factor 1/q! is removed here in the right-hand side of Eq.(A.4). 20

21 References [1] M.B. Green, J.H. Schwarz and E. Witten, Superstring Theory I,II (Cambridge Univ. Press, Cambridge, 1987); L. Brink and M. Henneaux, Principles of String Theory(Plenum Press, New York, 1988). [2] K. Kikkawa and M. Yamasaki, Prog. Theor. Phys.76 (1986) 1379; J. Hoppe, Elem. Part. Res. J. (Kyoto) 80 (1989) 145; M. Yamanobe, P-Branes in the Extended Picture of Elementary Particles (Ph.D thesis, Science Univ. of Tokyo, 1996); S. Ishikawa, Y. Iwama, T. Miyazaki and M. Yamanobe, Int. J. Mod. Phys. A10 (1995) 4671; S. Ishikawa, Y. Iwama, T. Miyazaki, K. Yamamoto, M. Yamanobe and R. Yoshida, Prog. Theor. Phys. 96 (1996) 227. [3] C.J. Isham, R. Penrose and P.W. Sciama(Editors), Quantum Gravity 2 : a Second Oxford Symposium (Clarendon Press, Oxford, 1981); F. David, Simplicial Quantum Gravity and Random Lattices, in Gravitation and Quantizations (Editors : B. Julia and J. Zinn-Justin, Les Houches 1992 Session LVII, pp , Elsevier Sci. B.V., 1995); P. Pi Francesco, P. Ginsparg and J. Zinn-Justin, Phys. Rep. 254 (1995) 1. [4] Y. Akizuki, Harmonic Integral, 2nd Edition (Iwanami, Tokyo, 1972). [5] We will also call the local coordinate system by the name of the manifold itself. [6] Our manifold is assumed to be compact, so that harmonicity reduces to Eq.(2.4). [7] We are transforming a local coordinate system into another; remember the footnote [5]. 21

22 [8] Here, and henceforth, the components of the tensors A i1 i 2...i n are always antisymmetric with respect to the exchange of suffices. [9] R.P. Feynman and J.A. Wheeler, Rev. Mod. Phys. 21 (1949) 425; M. Kalb and P. Ramond, Phys. Rev. D9 (1974) 2273; M. Yamanobe, See Ref.[2]. [10] P.A.M. Dirac, Proc. Roy. Soc. A133 (1931) 60; Phys. Rev. 74 (1948)